Teaching Multiplicative Thinking is Hard

Maya was in grade 2 (~7 yo) when she came back from her gymnastics class one Saturday visibly upset: her coach didn’t believe she had woken up at 8.30 a.m., yet managed to be in the downtown gym by 9 a.m. The coach knew she lived on the South Shore and thought it was impossible to make it in such a short time.

I wasn’t homeschooling at that time, but was doing math spontaneously with my kids whenever the opportunity would arise: most of the time as playful discussions triggered by an actual quantifiable event, sometimes as pure number games we’d invent. But this was not the usual ad-hoc game or challenge that we delighted in at the dinner table or in the car. Maya asked for my help, because she felt embarrassed in front of her team, and powerless in redeeming her credibility (even if her coach had apparently just joked lightly about it).

She had a solid understanding of multiplication, and in relation to it, some grasp of division; she had just learned the metric units of length. I had never talked to her deliberately about speed. However, our Civic had a digital speedometer showing numbers that – the children seemed to understand – were related to how fast we are going (they could feel that). So I decided to help her solve a mathematical problem to prove she was telling the truth; I had to do it on the spot, and rather fast, to work with her fired up motivation and to not lose the authenticity of the situation.

And from setting up the problem, to explaining, and finally writing it for communication, I found myself navigating a web of decisions and obstacles that underlined the non-triviality of even the most basic modelling for the purpose of teaching.

First, I had to use numbers that my daughter could work with: 10.7 km – the actual distance from our home to the gym – could not work because she didn’t know decimals; I set it at 12 km, rather than at 10, because it would fit better, I thought, multiplicatively, with the 60 min measure of time needed to conceptualize the speed. Then, I decided quickly that the speed must be 80 km/h, even if 60km/h would have worked to solve her practical question of arriving in time; this decision was rather didactical: 60 km/h would have been too trivial, even for a novel situation such as this one was for her. So the problem for my daughter to solve was: “How long does it takes to get from home to the gym at a speed of 80 km/h, if the distance between the two is 12 km?”

Then came the part of explaining what does a reading of 80 on the speedometer mean (Maya seemed to easily accept that 80 is a good representative of the many readings of the speedometer – I chose not to use the word average for now). I told her that it is the distance traveled in 1h, or 60 minutes, and she seemed to get it but when I reformulated the problem again as a proportionality situation she was stuck: “If it takes 60 minutes to travel 80 km, how long does it take to travel 12 km?” I had to improvise a few more problems with the same structure, but more relatable, involving steps and seconds, which she actually played out by walking and counting before arriving at the following critical insight: “I need to find how much time is needed for 1 km, then multiply it by 12 to find out how long the drive was.” It was also her idea that dividing 60 by 80 is the same as dividing 6 by 8: at this point I wasn't sure if she had she lost – or rather abandoned – the quantitative connection. Also, I was uncertain about whether she understood this relation multiplicatively, or rather just thought of ignoring a zero (would she have thought, for example that, 42 divided by 56 is the same as 6 divided by 8?), but I let it be in order to maintain the flow in the reasoning. Interestingly, she didn’t convert to seconds, which would have made for a division in whole numbers, but drew six circles, which, she said, represent clocks, and broke them into “quarters” to distribute over 8, getting three quarters as a result, which she wrote in words: “trois quarts.” I asked “How long is that?” instead of the more school-like “Of what?” or “What is the unit?”, and she answered “75 seconds”. I thought this came from thinking about money, and said it: “This is not money”, to which she answered “Oh, yeah! [laughing] It’s 45 minutes…No, wait! This is a minute clock [pointing to the circle], so it’s…45 seconds”. Multiplying by 12 and getting the final result of 9 minutes was straightforward, and, in hindsight, I thought that 11 km – the number closer to the real distance – would have worked quite easily also.

Finally, writing down the solution so that she could convincingly present it to her coach was a lesson in and of itself: she had to include her other assumptions (such as that there are just two lights with a stop time of 30 seconds, or that there was no traffic on a Saturday morning) and a timeline, with the wake-up time, and the time taken for breakfast and getting dressed specified. I encouraged writing the solution schematically, but intelligibly, and using mathematical symbols, in particular the equal sign, grammatically (e.g., she initially wrote 60 km = 80 minutes – which I corrected). 

Maya was very satisfied with the result, and was happy to be able to, in her words, “prove I’m right.”

I was left wondering about the unpredictability of such modelling of reality using mathematics with children, about the constraints posed by the child’s cognitive abilities and present state of mathematical knowledge, about walking the thin line between letting her own the explanation and telling too much, and about letting slide some learning opportunities for the sake of the flow of the explanation. I also questioned whether I stayed true to the quantitative approach to problem solving I was professing in my teaching and research in math education.